Obtaining limited diffraction beams with the wavelet transform

Generating limited diffraction (or nondiffracting theoretically) beams involves deriving special solutions to a homogeneous wave equation. Previous results have been derived by the Fourier and Laplace transforms. The authors use the wavelet transform to obtain a novel nondiffracting solution. It can be shown that this new solution is equivalent to the second derivative of Lu-Greenleaf´s (1992) zero-th order X wave or the first derivative of Donnelly´s Localized wave. The advantage of the wavelet beams is their localization property, that is, they have smaller sidelobes compared with the previous results. The magnitude decays as 1/r³ along the lateral (r) direction. Although the slowest decay is still 1/√r asymptotically, the sidelobes are reduced to about half those of the broadband X wave. The authors also show that this new nondiffracting beam can be realized as a limited diffraction beam with finite energy and finite aperture ultrasound transducers